PART 2b. PLANAR KINETICS OF LUMPED MASS SYSTEMS

Motion and Deformation of Mechanical Systems with 2 Degrees of Freedom (2DOF)

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Textbook Chapter 3.5 (Lectures 14-18)

Acronyms: M: mass, K: stiffness, C: damping, ODE: ordinary differential equation, EOM: equation of motion, SEP: static equilibrium position, DOF: degree of freedom, FBD: free body diagram, CE: characteristic equation, CME: Principle of Conservation of Mechanical Energy

 

 

Lecture

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Major Topics/WHAT YOU WILL LEARN

Recommended homework problems

14

EOMs for 2-DOF mechanical M-K-C systems including base excitation. Free body diagrams (FBDs), selection of coordinates, and establishment of forces for mechanical elements connecting masses. Application of Newtons 2nd Law. Expressing EOMS in matrix form. Derivation of nonlinear EOMs for double pendulum.

3.49, 3.48, 3.50

15

Eigenanalysis of free response of 2-DOF undamped mechanical systems. Fundamental response function and derivation of system characteristic equation (CE). Solving CE: Eigenvalues (natural frequencies) and eigenvectors (mode shapes) of system, physical interpretation of natural frequencies and mode shapes. Transformation of coordinates to modal space via modal matrix of eigenvectors and uncoupling of system EOMs.

3.52a-d, 3.53a-d, 3.54

16

Transient response of 2-DOF undamped mechanical systems. Prediction of system transient (free) response using modal (natural) coordinates. Transformation to physical space. Examples of analysis and interpreting solutions (system responses). Examples of systems with rigid body motions: interpretation of null or zero natural frequency. Insights into the analysis of systems with viscous damping: lightly damped systems and systems with proportional damping. The concept of modal damping ratio. Application to the analysis of a 2-DOF system response after collision

3.52e-f, 3.53e-f, 3.56

17

Transient response of 2- DOF M-K-C system with proportional damping. Example of usage of modal coordinates Collision problem

3.60, 3.61

18

Forced periodic response of 2-DOF undamped mechanical systems. Steady state solution to harmonic force excitation using (a) modal coordinates and (b) direct substitution. Interpretation of regimes of operation: below, around and above natural frequencies. Effect of excitation frequency on amplitude (amplification factor) and phase lag of steady-state response. Insights into the forced periodic response of 2-DOF systems with viscous damping

3.62, 3.63 (forget the transient solution due to initial conditions and solve for the steady-state solution). Also, repeat with modal damping ratios = 0.2

Appendix

Application: The vibration Absorber

 

 

Get all: Lectures 14-18

 

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